Optimal Reinsurance Maximising Dividends: An Infinite-Dimensional Optimisation Approach and Numerical Results
49 Pages Posted: 19 Dec 2023
Abstract
This study considers a discrete-time dynamic framework for designing optimal reinsurance contracts to maximize the lifetime dividends of an insurance company. First, we address the problem under a budget constraint and subsequently incorporate a solvency condition for the insurer. Surprisingly, both settings prove equivalent to solving static problems. Our results emphasize the necessity of a tailored dividend rule to accommodate the dynamic nature of the setup. Specifying a dividend rule adds complexity to the problem, whose solution requires solving an infinite-dimensional Lagrangian problem. We show that among all admissible reinsurance contracts, the optimal solutions for all problems are multi-layer contracts. However, the layered structure from the latter problem depends on Lagrangian multipliers. Despite this, we propose a linear programming problem to approximate its solution, particularly leveraging the barrier dividend rule. Numerical results are presented for the Value-at-Risk (VaR), Average Value-at-Risk (AVaR) and Glue Value-at-Risk (GlueVaR), with a claim that these findings are applicable to any distortion risk measure. The study not only provides theoretical insights into the optimal reinsurance contract design but also offers a practical linear programming algorithm for solution approximation in real-world scenarios.
Keywords: Optimal reinsurance, Distortion risk measures, Dividends, Dynamic problem, Linear programming
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